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Theorem efival 12432
Description: The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
efival  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) ) )

Proof of Theorem efival
StepHypRef Expression
1 ax-icn 8796 . . . . . 6  |-  _i  e.  CC
2 mulcl 8821 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 651 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efcl 12364 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
53, 4syl 15 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
61negcli 9114 . . . . . 6  |-  -u _i  e.  CC
7 mulcl 8821 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
86, 7mpan 651 . . . . 5  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
9 efcl 12364 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
108, 9syl 15 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
115, 10addcld 8854 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
125, 10subcld 9157 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
13 2cn 9816 . . . . 5  |-  2  e.  CC
14 2ne0 9829 . . . . 5  |-  2  =/=  0
1513, 14pm3.2i 441 . . . 4  |-  ( 2  e.  CC  /\  2  =/=  0 )
16 divdir 9447 . . . 4  |-  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  2 )  =  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  +  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
1715, 16mp3an3 1266 . . 3  |-  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  2 )  =  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  +  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
1811, 12, 17syl2anc 642 . 2  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  / 
2 )  =  ( ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  +  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2
) ) )
1910, 5pncan3d 9160 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( exp `  (
_i  x.  A )
) )
2019oveq2d 5874 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( ( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
215, 10, 12addassd 8857 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( ( exp `  ( _i  x.  A ) )  +  ( ( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
2252timesd 9954 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
2320, 21, 223eqtr4d 2325 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( 2  x.  ( exp `  (
_i  x.  A )
) ) )
2423oveq1d 5873 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  / 
2 )  =  ( ( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 ) )
25 divcan3 9448 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
2613, 14, 25mp3an23 1269 . . . 4  |-  ( ( exp `  ( _i  x.  A ) )  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
275, 26syl 15 . . 3  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
2824, 27eqtr2d 2316 . 2  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  2 ) )
29 cosval 12403 . . 3  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
3013, 1mulcli 8842 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
31 ine0 9215 . . . . . . . 8  |-  _i  =/=  0
3213, 1, 14, 31mulne0i 9411 . . . . . . 7  |-  ( 2  x.  _i )  =/=  0
3330, 32pm3.2i 441 . . . . . 6  |-  ( ( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 )
34 div12 9446 . . . . . 6  |-  ( ( _i  e.  CC  /\  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 ) )  ->  ( _i  x.  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  (
_i  /  ( 2  x.  _i ) ) ) )
351, 33, 34mp3an13 1268 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  ->  (
_i  x.  ( (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  (
_i  /  ( 2  x.  _i ) ) ) )
3612, 35syl 15 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  (
_i  /  ( 2  x.  _i ) ) ) )
37 sinval 12402 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
3837oveq2d 5874 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( _i  x.  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) ) )
39 divrec 9440 . . . . . . 7  |-  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) ) )
4013, 14, 39mp3an23 1269 . . . . . 6  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) ) )
4112, 40syl 15 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) ) )
421mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  _i )  =  _i
4342oveq1i 5868 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( _i  /  (
2  x.  _i ) )
441, 31dividi 9493 . . . . . . . . . 10  |-  ( _i 
/  _i )  =  1
4544oveq2i 5869 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  /  2
)  x.  1 )
46 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
4746, 13, 1, 1, 14, 31divmuldivi 9520 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )
4845, 47eqtr3i 2305 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( ( 1  x.  _i )  /  (
2  x.  _i ) )
4913, 14reccli 9490 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
5049mulid1i 8839 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( 1  /  2
)
5148, 50eqtr3i 2305 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( 1  /  2
)
5243, 51eqtr3i 2305 . . . . . 6  |-  ( _i 
/  ( 2  x.  _i ) )  =  ( 1  /  2
)
5352oveq2i 5869 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( _i  / 
( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) )
5441, 53syl6eqr 2333 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( _i  / 
( 2  x.  _i ) ) ) )
5536, 38, 543eqtr4d 2325 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5629, 55oveq12d 5876 . 2  |-  ( A  e.  CC  ->  (
( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) )  =  ( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2
)  +  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
5718, 28, 563eqtr4d 2325 1  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   expce 12343   sincsin 12345   cosccos 12346
This theorem is referenced by:  efmival  12433  efeul  12442  efieq  12443  sinadd  12444  cosadd  12445  absefi  12476  demoivre  12480  efhalfpi  19839  efipi  19841  ef2pi  19845  efimpi  19859  efif1olem4  19907  1cubrlem  20137  asinsin  20188  atantan  20219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352
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